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MATH 148-MIDTERM, SPRING 2019

Please return the exam in class at 9:40am on May 07. You may only consult all the resources that we posted at iLearn and your notes. You must work independently. In particular, you may not work with others. You may not email your instructor or TA for math quesitons during the exam time.

Please note that most of the exam problems are open. Hence, highly simiar solutions of exam problems are generally not possible and can be easily detected. In particular, unreasonably highly similar exam copies will be reported as cheating case to the Student Conduct & Academic Integrity Programs.

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2 MATH 148-MIDTERM, SPRING 2019

Problems:

1. (5 pts) Translate the following statement into the language of ODE theories (i.e. theories concerning existence and uniqueness, global exsitence, of solutions of IVP, etc.):

Consider a system determined by some ODE system. Trajectories exist for all initial states. Moreover they exist for all time. However, for some choice of the initial states, their the future are uniquely determined while for their past, there are infinitely many possibilities.

Can such a ODE system define a dynamical system? Why?

2. (7 pts) Construct a one-dimension dynamical systems x′ = F (x) so that it has three equilibrium points. Moreover, two of them are semi-stable and the third is a sink which sits in-between the two semi-stable ones. Sketch its phase line.

3. (7 pts) Construct a planar nonlinear system with the following properties:

(1) the origin ~0 is an equlibrium point and the circle x2 + y2 = 4 corresponds to a cycle (i.e. a periodic solution).

(2) ~0 is a spiral sink and the cycle corresponds to x2 + y2 = 4 repels all nearby trajectories. Precisely, as t → ∞, all other solutions (than the periodic solution and the equilibrium point) move away from the circle x2 + y2 = 4.

(3) all trajectories other than ~0 rotate clockwisely.

Then sketch the phase portraits.

4. (9 pts) Construct a one-parameter family of planar linear dynamical systems defined by

~x′ = Aa~x, a ∈ I, where I ⊂ R is some interval, with the following properties:

(1) there is an unique value of the parameter, a0 ∈ I, where a bifurcation occurs. Moreover,

(2) for a ∈ I with a < a0, ~0 is a saddle and for a ∈ I with a > a0, ~0 is a sink. Sketch the phase portraits for any two values of a, a1 and a2, such that a1 < a0 < a2. Briefly discuss that what happens at a = a0 (you don’t need to sketch the phase portraits for a = a0).

5. (7 pts) Let ΦA and ΦB : R × R2 → R2 be two planar linear dynamical systems defined by ~x′ = A~x and ~x′ = B~x, respectively. Assume that both A and B have no zero eigenvalues. Suppose ΦA is conjugate to ΦB via a homeomorphism h : R2 → R2. First show that it must happen that h(~0) = ~0. Then show that ~0 is a center point of ΦA if and only if it’s a center point of ΦB.